Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Factorization of polynomials over finite fields wikipedia , lookup
Polynomial ring wikipedia , lookup
Quartic function wikipedia , lookup
System of polynomial equations wikipedia , lookup
Eisenstein's criterion wikipedia , lookup
Dessin d'enfant wikipedia , lookup
Signal-flow graph wikipedia , lookup
Pre-Test 3 Chapters 4, 5 Section 4.1 1. Math 141 1 May 28, 2014 2. Math 141 2 May 28, 2014 Section 4.2 3. Determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line. (a) x 1 5 10 12 f ( x) 8 16 26 32 Nonlinear (b) x 1 2 4 5 f ( x) 3 5 9 11 5 3 9 5 11 9 2 linear 2 1 4 2 5 4 (1,3) : 3 2(1) b; b 1, y 2 x 1 m 4. For the given data points below use the calculator to find the best fit line. x 6 8 10 12 f ( x) 1 4 6 10 y 1.45 x 7.8 Section 4.3 5. For the quadratic equation f ( x) 2 x2 12 x 22 , find the following: a. Does it open up or down? b. Find the vertex. Math 141 3 May 28, 2014 c. d. e. f. Graph the function. Find the y-intercept and x-intercept(s). Find the domain and range. Determine where the function is increasing and decreasing. (a) Up (b) b (12) 12 3, 2a 2(2) 4 f (3) 4; (3, 4) (c) y 9 8 7 6 5 4 (3,4) 3 2 1 x 1 2 3 4 5 (d) f(0) = 22; y-intercept: (0,22); No x-intercepts (e) Domain: All Reals, Range: y ≥ 4 (f) Increasing: (3, ∞); Decreasing: (-∞, 3) 6. Determine algebraically, without graphing, the minimum value of the quadratic function f ( x) 2 x 2 3x 2 . b (3) 3 .75, 2a 2(2) 4 f (.75) .875; (.75, .875) Section 4.4 7. Maximizing Revenue The price p ( in dollars) and the quantity x sold of a certain product obey the demand equation p 1 x 15 10 (a) Express the revenue R as a function of x. 1 1 2 R( x) xp x x 15 x 15 x 10 10 Math 141 4 May 28, 2014 (b) What is the revenue if 20 units are sold? R(20) 1 (20) 2 15(20) 260 10 (c) What quantity x maximizes revenue? What is the maximum revenue? b 15 75 2a 1 2 10 1 2 R(75) 75 15(75) $562.50 10 (d) What price should the company charge to maximize revenue? p 1 (75) 15 $7.50 10 Section 5.1 1. Determine which functions are polynomial functions. For those that are, state the degree. (a) f(x) = 5x2 + 4x4 Yes, degree 4 (b) F(x) = x2 5 x3 No 2. Form a polynomial function whose real zeros and degree are given. a. Zeros: -2, 3,4; degree 3 f(x) = (x + 2)(x – 3)(x – 4) = ( x 2 x 6)( x 4) x3 5x 2 2 x 24 b. Zeros: -1, multiplicity 1; 3, multiplicity 2; degree 3 f ( x) ( x 1)( x 3)2 ( x 1)( x 2 6 x 9) x3 5 x 2 3x 9 3. For the polynomial function f(x) = 4(x + 4)(x + 3)3: (a) List each real zero and its multiplicity (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Determine the maximum number of turning points for the graph. Math 141 5 May 28, 2014 (d) Determine the end behavior; that is, find the power function that the graph resembles for large values of |x|. 4. Analyze the polynomial function f(x) = (x – 1)(x + 3)2 by finding the following: (a) Determine the end behavior of the graph of the function. (b) Find the x- and y-intercepts of the graph of the function. (c) Determine the zeros of the function and their multiplicity. (d) Use a graphing calculator to graph the function. (e) Find the turning points of the graph. (f) Find the domain and range of the function. (g) Use the graph to determine where the function is increasing and where it is decreasing. Math 141 6 May 28, 2014 Section 5.2 5. Find the remainder when f ( x) 3x 4 6 x3 5x 10 is divided by x - 2. Then use the Factor Theorem to determine whether x - 2 is a factor of f(x). 6. Determine the maximum number of real zeros that f ( x) x5 x 4 2 x 2 3 may have. Then list the potential rational zeros of the function. Do not attempt to find the zeros. Math 141 7 May 28, 2014 7. Find the bounds to the zeros of each polynomial function. a. b. 8. Find the real zeros of x x 2 x 4 x 8 synthetic division to factor the function. 4 Math 141 3 2 8 0. Use the real zeros and May 28, 2014 Section 5.3 9. Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. a. Degree 3; zeros: 4, 3 + i 3 - i b. Degree; 6 zeros: 2, 2 + i, -3 – i, 0 2 – i -3 + i Math 141 9 May 28, 2014 10. Form a polynomial f(x) with real coefficients having the following degree and zeros. a. Degree 4; zeros: 2 + i, and 3, multiplicity 2 f ( x) ( x (2 i))( x (2 i))( x 3)( x 3) f ( x) ( x (2 i))( x (2 i))( x 2 6 x 9) f ( x) ( x 2 i ))( x 2 i ))( x 2 6 x 9) f ( x) ( x 2 2 x xi 2 x 4 2i ix 2i i 2 )( x 2 6 x 9) f ( x) ( x 2 4 x 5)( x 2 6 x 9) f ( x) x 4 6 x 3 9 x 2 4 x 3 24 x 2 36 x 5 x 2 30 x 45 f ( x) x 4 10 x 3 38 x 2 66 x 45 Section 5.4 11. Find the domain of each rational function. (a) R( x) 5x2 x3 (b) Q( x) x(1 x) 3x 2 5 x 2 12. Find the vertical, horizontal, and oblique asymptotes, if any, for each rational function. (a) R( x) Math 141 3x 5 x6 (b) F ( x) x2 6 x 5 2x2 7 x 5 10 May 28, 2014 Section 5.5 13. Analyze the rational functions below by finding the following: R( x) a. b. c. d. e. f. g. h. Math 141 ( x 1)( x 2 x 1) x G ( x) x2 2 x ( x 1)( x 2) Factor the numerator and denominator. Find the domain. Write the function in lowest terms. Find the x- and y-intercepts of the graph. Test for symmetry. Is it symmetry with respect to the y-axis, origin, or neither. Local the vertical asymptotes. Local the horizontal or oblique asymptotes, if any. Graph the function using a graphing utility. Draw a complete graph of the function by hand using the information from steps a – g. 11 May 28, 2014 14. Drug Concentration The concentration C of a certain drug in a patient’s 50t bloodstream t minutes after injections is given by C (t ) 2 . t 25 a. Find the horizontal asymptote of C(t). What happens to the concentration of the drug as t increases? b. Using your graphing utility, graph C = C(t). c. Determine the time at which the concentration is highest. Math 141 12 May 28, 2014